Intertwined supply network design under facility and transportation disruption from the viability perspective

To ensure viability, it is necessary for an intertwined supply network (ISN) system to optimise network structure to provide flexible redundancy in response to changing environment. Considering resilience methods are usually designed as reactions to single discrete disruptions rather than situational reactions to real-time changes, this study proposes a novel redundancy optimisation approach dynamically providing each demand market with a pair of supply routes to optimise the flexible redundancy of ISN, thereby ensuring the survivability of supply chains and demand markets under continuous changes. Based on this, we propose the ISN design (ISND) model to capture the trade-off between total cost and viability performance under facility and transportation disruption. The Lagrangian relaxation algorithm, combined with the sub-gradient method and the improved cellular genetic algorithm, are utilised for solving problems of different scales. To test the performance of the model and corresponding algorithms, we also conduct a numerical analysis of the data from medical equipment ISN in southern China. The results indicate that the ISND model can effectively optimise ISN structures, which makes it possible to dynamically provide flexible redundancy; the two algorithms also show good calculation efficiency. The relationship between ISN structure and viability performance is thus observed and explained.


Introduction
As a new instigator of disruptions, the coronavirus 2019 (COVID-19) pandemic has significantly affected the global and local economies as well as supply chains (SCs) (Ivanov 2020a). Since January 2020, the number of COVID-19 cases has increased exponentially in Asia, Europe, and the USA, resulting in the implementation of border closures and quarantines. Some factories and transportation links became unavailable under this long-term global disruption, with the resulting material shortages and delivery delays propagating downstream SCs and causing a ripple effect and performance degradation in terms of revenue, service level, and productivity (Choi 2021;Dolgui, Ivanov, and Rozhkov 2020;Li et al. 2020;Dolgui, Ivanov, and Sokolov 2018;Wu, Blackhurst, and O'grady 2007;Blackhurst et al. 2005). Some SCs witnessed a drastic surge in demand but the supply could not cope with this situation (e.g. facial masks, hand sanitisers). As such, the question of market and society survivability emerged. For other SCs, the demand and supply have dropped drastically, resulting CONTACT Jianming Yao jmyao@163.com Supplemental data for this article can be accessed here. https://doi.org/10. 1080/00207543.2021.1930237 in production stops (e.g. automotive industry), which posed the danger of bankruptcy for these SCs (Yoon et al. 2018). As pointed by Ivanov and Dolgui (2020), under the current long-term pandemic situation, these questions go beyond the existing SC resilience concept, since they cannot be resolved from a narrow SC perspective. Reallife SCs do not usually operate autonomously but span and interconnect within and even across business sectors, forming sophisticated intertwined supply networks (ISN). An ISN can be defined as the entirety of interconnected SCs which, in their integrity secure the goods and services for the society and markets. The ISNs are systems with structural dynamics, in which the firms may exhibit multiple behaviours, such as changing their positions and roles in interconnected or competing SCs, and sharing various resources . However, resilience analysis is usually related to the level of individual SCs or supply chain networks (SCNs) with star-like or tier-based structures in a closed system setting Dixit, Verma, and Tiwari 2020;Fattahi, Govindan, and Maihami 2020), which indicates both unfit and insufficient for sophisticated ISNs.
Therefore, under the current long-term and unpredictable global disruptions, a perspective at larger scales considering interconnected complex supply networks and production systems is required urgently. Recently, the viability perspective, spanning the resilience, adaptability, and sustainability in complex systems, has been introduced into ISN management by Ivanov and Dolgui (2020). Viability is a concept stemming from ecology, biological systems (Aubin 1991), and cybernetics (Beer 1981), which describes a system's ability to meet the demand of surviving in a changing environment. Compared to resilience, viability is a behaviour-driven property of a system with structural dynamics, which considers system evolution through disruption reaction balancing in an open system context. Viability analyses are survival-oriented without fixed time window in a long-term scale . Under the COVID-19 pandemic, the viability perspective is necessary to ensure the sustainability of ISNs and demand markets. A viable ISN can survive at the times of long-term, global disruptions by adjusting capacities utilizations and their allocations to demands in response to internal and external changes in line with the sustainable developments to secure the provision of society and markets with goods and services in long-term perspective Ivanov 2020b).
The global COVID-19 pandemic and SC collapse move the ISN and market survivability issues to the forefront of risk management discussions (Keogh 2020). Some proactive resilience methods such as holding safe inventories, utilising backup supply and transportation infrastructures are less efficient in ensuring long-term survivability in changing environment . As such, to ensure ISN viability, it is necessary to optimise network structure to dynamically create flexible redundancy in response to the real-time changes caused by long-term and unpredictable global disruptions (Li et al. 2020;Ivanov, Das, and Choi 2018). In this study, we consider ISN disruption risks resulting from facilities and transportation links.
Motivated by the study of Ivanov and Dolgui in 2019, which indicated that SCNs' structure properties played an important role in maintaining reliability and achieving resilience, we propose a novel structural redundancy optimisation approach, which provides each customer with a pair of supply routes, thereby securing service/product supply and maintaining ISN survivability. Each supply route consists of facilities at different levels of the network and transportation links. Customers are primarily served by the main supply routes, while backup supply routes are used when the main supply routes fail.
All supply routes are organised and designed dynamically utilising the available facilities and transportation links according to the external environment changes.
Based on the above-mentioned redundancy optimisation approach, the ISN design (ISND) model is proposed to optimise the ISN structure to ensure viability performance at an optimal cost. An effective adaptive mechanism is constructed correspondingly, which makes it feasible for the decision makers to adjust and redesign ISN structure dynamically, thus handling environment changes and uncertainties. To improve the recovery efficiency and ISNs' ability to dynamically adapt to external continuous changes, the ISND model removes the single assignment constraints typically used in the existing SCN design (SCND) models (Ramshani et al. 2019;Gendron, Khuong, and Semet 2016). Moreover, the ISND model considers the independent disruption risks resulting from facilities and transportation failures in more than one level of the network, which are overlooked in some SCND studies, but are necessary and important in case of long-term and unpredictable large-scale global disruptions.
We take the medical equipment ISN (me-ISN) in southern China as an example to analyse ISND model's performance in terms of improving ISN viability and reducing economic loss in the context of the current COVID-19 pandemic. The me-ISN is an entirety of interconnected SCs which, in their integrity secure the provision of southern China's medical market with equipment such as facial masks, personal protective equipment, and respirators. We simplify the me-ISN into a three-level network, including multiple suppliers, focal firms, and demand markets, where independent disruption risks exist in each level of the network at the same time. To ensure viability while controlling the operation cost in a continuously changing environment, the ISND model and its adaptive mechanism are applied to dynamically optimise the structure of the sophisticated me-ISN. By analysing the properties of the ISND model, the Lagrangian relaxation (LR) algorithm and cellular genetic algorithm are then utilised to solve problems at different scales. The results show that the disruptions caused by COVID-19 are unavoidable, but the negative effects can be mitigated by optimising the ISN structure to dynamically provide flexible redundancy. We also find that the disruption levels and maximum supply routes have positive effects on the total cost of the me-ISN design. Moreover, the relationship between ISN structure and viability performance is observed and explained.
The remainder of this paper is structured as follows. Section 2 reviews the literature on SCND under disruptions, ISN, and viability. The ISND model, characterised by dynamic alternations of main routes and backup routes considering facility and transportation disruptions, is proposed and simplified in Section 3. In Sections 4 and 5, we utilise the LR algorithm and improved cellular genetic algorithm to solve the ISND model. Section 6 illustrates the effectiveness and performances of the proposed ISND model and its algorithm through a numerical study based on me-ISN. We conclude and provide future research directions in Section 7.

Literature review
Here, we introduce two literature streams that are relevant to this study: (i) SCND under disruption risks and resilience methods and (ii) ISN and viability.

Supply chain network design under disruptions
Many real-life service facilities and transportation links are subject to probabilistic disruptions caused by natural disasters such as earthquakes and epidemics; and human actions such as terror attacks, cyber-attacks, and traffic congestion (Snyder and Daskin 2005;Snyder et al. 2016). In today's global turbulent environment, disruptions can occur at any time and echelon in a SCN, which is why ignoring their possibility could lead to a suboptimal design that is vulnerable to even infrequent disruptions (Li and Ouyang 2010).
In supply chain network design (SCND) field, the risk of uncertain disruption including facility disruption and transportation disruption has received much attention. The major part of these studies usually aim to select a set of facilities and transportation links to design a reliable SCN, with the objectives to minimise total cost and alleviate negative impacts caused by disruptions (Schmitt et al. 2017;Baghalian, Rezapour, and Farahani 2013;Peng et al. 2011), make a trade-off between resilience and environment impacts (Mohammed et al. 2021;Mohammed 2020), or make a balance between transportation time and total cost (Diabat, Jabbarzadeh, and Khosrojerdi 2019;Fattahi, Govindan, and Keyvanshokooh 2017), etc.
Multiple modelling approaches have been applied to SCND problem under the risk of uncertainty. Most studies adopt models which assume that a risk-neutral decision maker who wishes to optimise the expected value of the objective function. For example, Snyder and Daskin (2005) proposed reliability models based on a p-median problem and an uncapacitated fixed-charge location problem, in which facilities are subject to disruptions. Their models aimed to minimise facility location costs, while considering the expected transportation cost when an unexpected disruption occurs. Further, Jahani et al. (2018) proposed a stochastic model to redesign a SCN when the existing logistics no longer optimally met the objectives of the enterprise, with the purpose to maximise expected profits. In their model, the uncertainty in demand and price, and the correlation structure between them have been introduced into the optimisation model for the first time. Their work was extended by Jahani, Abbasi, and Talluri (2019) by integrating the common operational SCND considerations with the financial concerns expressed by creditors, investors, and stockholders. Moreover, Mohammed et al. (2019) developed a green and resilient fuzzy multi-objective programming model for a G-resilient SCND to minimise the expected total cost and environmental impact, and maximise the value of resilience pillars. Besides, some studies focus on risk aversion decision making through bi-level model formulation and optimising worst-case objectives (Losada, Scaparra, and O'Hanley 2012;Medal, Pohl, and Rossetti 2014).
There are also studies that consider the risk preferences of decision makers through scenario-based models. For instance, Peng et al. (2011) formulated a scenariobased SCND model to minimise the total cost under normal conditions while reducing the disruption risk using the p-robustness criterion. A hybrid meta-heuristic algorithm was proposed to solve this model. Baghalian, Rezapour, and Farahani (2013) developed a scenariobased model for designing a SCN with objective to maximise profit under the risks of disruption. They formulated the problem using mixed integer nonlinear programming and approximated it using multiple linear regressions.
In most of the SCND studies with consideration of the disruption occurrence probability, the facility and transportation disruption are usually considered as random events, and the disruption probability during a certain period is usually treated as a known distribution assumed to be accurately predictable based on historical data or by using disaster-specific sample scenarios that rely on specific disaster models (e.g. earthquake or hurricane models) (Yan and Ji 2020). However, the above methods are not applicable to some situations when historical data are scant and lowly informative. For example, under long-term unpredictable crises such as the COVID-19 pandemic, disruptive events that are quite different in scale, complexity, severity, and duration might occur in any echelon of SCN at any time. Therefore, both randomness (stochastic variability of all possible outcomes of a situation) and fuzziness (incomplete or imprecise knowledge regarding the situation) exist in the estimation process of disruption risks (Wang, Herty, and Zhao 2016).
Since this study focuses on ISN structural optimisation problem under facility and transportation disruptions, the probability of disruption occurrence is an important reference for decision making process that should be taken into the model. Therefore, the fuzzy random theory is introduced to overcome the problem of inability to acquire enough valid data to estimate the disruption probability. Accordingly, disruption events are characterised by a belief degree based on experts' estimations, thus avoiding dependence on perfect historical data (Liu and Liu 2003;Puri and Ralescu 1986;Kwakernaak 1978). The fuzzy random theory has already been applied to the SCND problem without considering disruption risks. For example, Wen and Kang (2011) discussed the location and allocation problem under fuzzy random demands. Moreover, a fuzzy random facility location model with fuzzy random costs and demand was built by Wang and Watada (2012) to minimise the value-at-risk of investment by determining the optimal locations as well as the capacities of the new facilities to open with a hybrid modified particle swarm.

Supply chain resilience methods
Since SC and SCNs are confronted with numerous events that threaten to disrupt SCs' operational activities and jeopardise efficient and effective performance (Ivanov and Das 2020), SC and SCN resilience has gained increasing attention. Resilience describes the ability to withstand a disruption or a series of disruptions and then recover performance (Fattahi and Govindan 2018;Spiegler, Naim, and Wikner 2012). Ponomarov and Holcomb (2009) identified three phases of SC resilience: readiness, response, and recovery. Similarly, Kamalahmadi and Parast (2017) identified four phases of SC resilience: anticipation, resistance, recovery, and response. Accordingly, two critical capacities of resilience can be recognised: (1) the resistance capacity, which describes the ability of a system to minimise the impact of a disruption by evading a hazard entirely or reacting early, and (2) recovery capacity, which is the ability of a system to return to a steady state of operational capacity after a disruption (Tan, Zhang, and Cai 2019). Correspondingly, there are two broad categories of resilience methods: proactive and reactive. Hosseini, Ivanov, and Dolgui (2019) pointed out the three defence lines against disruptions, which can be described by three resilience capacities, including absorptive, adaptive and restorative capacities. At the pre-disruption stage, the methods to improve the absorptive capacity are emphasised, including supplier segregation (Hosseini and Barker 2016), multiple-sourcing strategy (Yildiz et al. 2016;Ivanov 2018), inventory positioning (Chopra and Sodhi 2004;Turnquist and Vugrin 2013), and multiple transportation channels (Khalili, Jolai, and Torabi 2016). At the post-disruption stage, scholars explored the methods to improve the adaptive capacity, which includes backup suppliers (Jabarzadeh, Fahimnia, and Sabouhi 2018;Tomlin 2006), rerouting (Khaled et al. 2015;Wang, Herty, and Zhao 2016) and communication and substitution (Mancheri et al. 2018), and the restorative capacity (Hosseini and Barker 2016).
As is known to us, creating redundancy is a way to achieve resilience across an SC/SCN (Kamalahmadi and Parast 2017). The redundancy allows the SCs and SCNs to maintain the production process even when some facilities and transportation links are disrupted and provides backup resources that can be utilised during disruptions. Among the aforementioned resilience strategies and methods, the following ones can be linked to redundancy creation: (1) Holding an emergency stock of materials and finished goods in some facilities if disruption occurs; (2) Having backup suppliers to allow the SCN to maintain the production process even if one supplier is down during the disruption; (3) Utilising the backup facilities of a critical facility during disruptions to maintain the production process; and (4) Protecting suppliers against disruptions by increasing the redundancy in operations.
The existing studies on resilience have identified the relationship between resilience and SCN structures. The structural properties of SCN play a critical role in maintaining reliability and achieving resilience (Tan, Cai, and Zhang 2020;Bode and Wagner 2015;Dubey et al. 2019a). Ivanov and Dolgui (2019) emphasised that complex networks become more vulnerable to severe disruptions, which changes SC structures and is involved with SC structural dynamics. As SCN structures serve such an important role in disruption resistance, the optimisation of flexible redundancy should start with the optimisation of SCN structures.
In summary, the SCND problem towards resilience under the risk of uncertain disruption, especially the facility disruption, has been well studied in many contexts. However, the long-term independent disruption risks resulting from facilities and transportation failures in more than one level of SCN simultaneously have not been explicitly captured in existing studies. Under the long-term disruptions such as COVID-19, facilities and transportation links at different levels of SCN are prone to disruptions with independent probability, and the disruptions may occur at different levels at the same time. It is necessary to take the independent disruption risks resulting from facilities and transportation failures in more than one level of SCN into consideration. Besides, single assignment constraints are common in some SCND models, which reduces difficulties in solving model but imposes additional restrictions (Ramshani et al. 2019;Gendron, Khuong, and Semet 2016). The constraints required that each level should be served by only one facility on its previous level, for example, one supplier for one focal firm, and one focal firm for one demand market, which may lead to delay in reassignment after disruption due to the lack of flexible redundancy. To ensure immediate response to disruptions, it is necessary to consider the construction of optimisation model without single assignment constraints.
In this study, we consider the independent disruption risks resulting from facilities and transportation failures in more than one level of the ISN and propose a novel redundancy optimisation approach, which is characterised by dynamically providing each demand market with a pair of supply routes. When disruptions occur, customers can be reassigned to the other supply route within a short period, which can effectively alleviate the survivability risk and economic loss resulting from the delayed response under a changing environment. To the best of our knowledge, few existing studies consider redundancy creation by providing backup supply routes.

Intertwined supply network
In practice, SCs do not usually operate autonomously but span and interconnect within and even across business sectors, forming supply ecosystems or ISNs . There are three main properties that distinguish ISNs from linearly directed SCNs with static structures: (1) Open system: Cross-industry intertwining interconnection is common for ISNs. (2) Structural dynamics: ISNs consist of different structures-business processes and technological, organisational, technical, topological, informational, and financial structures-which are interrelated and change in their dynamics. For example, all participants involved in ISNs may exhibit multiple behaviours, such as changing the buyer-supplier roles in interconnected or even competing SCs, which makes the SCN structure more dynamic and flexible.
(3) Integrity: The ISN is an entirety of interconnected SCs which, in their integrity, secure the provision of the society and markets with services or products (e.g. food service, mobility service or communication service).
The rich content of ISNs is derived from three research streams: collaborative networks, complex adaptive system, and SC structural dynamics. In the collaborative networks field, scholars have pointed out that the collaboration of SCs enhanced the preparedness of participants to promptly form virtual organisations and enabled them to successfully tender for large scale and distributed projects (Dolgui and Proth 2010;Shamsuzzoha and Helo 2017;Hernández et al. 2014;Jayaram and Pathak 2013;Fornasiero and Zangiacomi 2013;Ivanov and Sokolov 2012;Noran 2009).
From the perspective of adaptive systems, collaboration between SCs usually leads to the formation of complex multi-structure collaboration networks, which can be referred to as complex systems. Therefore, the complex adaptive system perspective can be applied to provide guidelines on how SCNs with complex organisational structures and functions develop and which organisations and functionality are attainable (Choi, Dooley, and Rungtusanatham 2001;Surana et al. 2005). Furthermore, since the organisational structure and operational features of SCN constitute a major portion of the research on coping with SC disruption risk, adaptive collaboration can improve SCNs' capacity to cope with disruption risks .
Regarding SC structural dynamics, Ivanov, Sokolov, and Kaeschel (2010) proposed a new conceptual framework for the multi-structural planning and operations of adaptive SCs under structural dynamics considerations. Further, Chen (2017) suggested that enterprises should build and manage product/service customisation and dynamic structural collaborative networks to respond to market demand flexibly with competitive price and high product/service quality. These studies enhanced the managerial insights into advanced SC management, making SCs more agile, flexible, and responsive.
It is common to see SCs which are spanned and interconnected within and even across the business sectors. Through analysing the existing studies on ISN, few of them considered the ISN design optimisation problems based on its structural dynamics. Through analysing the ISN characteristics, this study focuses on the dynamic structure optimisation of ISNs, which enriches the research on complex adaptive system and structural dynamics.
The implementation of the redundancy optimisation approach proposed in this study is based on the structural characteristics of ISNs. As SCs usually connect, intersect, and intertwine with other SCs from different enterprises and industries in ISN, thus focal firms can exhibit multiple behaviours, such as changing their positions and roles in interconnected or competing SCs, and sharing various resources (e.g. inventory, warehouses) (Zhao, Zuo, and Blackhurst 2019;Fraccascia, Giannoccaro, and Albino 2017;Dubey, Gunasekaran, and Papadopoulos 2019b).
The structural dynamics of ISN enable facility and transportation links from different SCs to create alternating supply routes that respectively serve as main and backup supply routes at different periods, which makes flexible redundancy possible.

Viability perspective
As pointed in Section 2.1, most real-world SCNs are subject to disruption risks caused by natural disasters and human actions, which makes resilience particularly important. However, with the advent of information technology, SCNs have acquired a complexity almost equivalent to that of biological systems, especially for ISNs. However, ISNs' survivability problems go beyond the concept of resilience because resilience is usually related to the levels of individual SCs or SCNs. Since these problems cannot be solved from the perspective of narrow single SCs, a higher-level perspective focusing on larger-scale analysis is urgently necessary.
Therefore, the perspective of viability has been introduced into ISN management  from ecology, biological systems (Aubin 1991), and cybernetics (Beer 1981). In sum, viability describes a system's ability to meet the demands of surviving in a changing environment. Ivanov (2020b) pointed out the two main objectives of viability under long-term, unpredictable crises such as the COVID-19 pandemic, which are (1) maintaining the ISN survivability in the long term; and (2) securing the provision of a society with products or services.
As summarised by Ivanov and Dolgui (2020), viability is a behaviour-driven property (continuous system change) of an open system with structural dynamics, which considers system evolution through disruption reaction balancing. Viability analysis is survivaloriented without fixed time windows for the long-term scale, while resilience is considered to be a disruptiondriven SC property (single, discrete, unique events) within a closed system setting, which relates to singular disruption-reaction analyses. As such, resilience analyses are performance-oriented for some fixed time windows and mostly linear, single-flow directed SC systems. It is recognised that viability can extend the resilience angle toward survivability in case of extraordinary events (Ivanov and Das 2020).
Nowadays, most SCND studies under disruptions of facilities or transportation are conducted within resilience perspective. SCNs are usually redesigned and strengthened to mitigate the negative impacts resulting from disruptions and improve economic performance within a fixed time-window. However, the resilience perspective is not applicable for long-term unpredictable global disruptions, such as the COVID-19 pandemic.
As pointed by Dolgui (2020, 2021), under such 'super uncertainty', disruptive events that are quite different in scale, complexity, and severity can occur at any echelon of the supply network at any time, which can radically change operational conditions over long periods.
The example of the COVID-19 outbreak shows that, in case of extreme events, SCs' resistance to disruptions needs to be considered at the viability levels to avoid SC and market collapse and secure the provision of goods and services. Therefore, in this study, when designing a network structure under a long-term unpredictable crisis, we consider not only the mitigation of negative impacts and the improvement of economic performance, but also the survivability of ISN system and demand market in continuously changing environment (Hosseini, Ivanov, and Blackhurst 2020). Table 1 summarises the characteristics of existing SCND models under the risk of disruptions.

Research gaps
Through analysing previous studies, we identify the following four research gaps in literature.
(1) Since the resilience perspective is not applicable for long-term unpredictable global disruptions, the viability perspective which focuses on situational reactions to real-time changes rather than reactions to single, discrete disruptions has been introduced into ISN management to handle those super crises. Following the work of Ivanov and Dolgui (2020), this study makes a preliminary exploration on network redundancy optimisation from the viability perspective. The ISND model and the corresponding adaptive mechanism are proposed to optimise the structure of ISNs through a trade-off between operation cost and viability performance. (2) Some common disruption probability estimation methods used in exiting SCND studies are not applicable to those practical problems when historical data are scant and lowly informative. In this study, the fuzzy random theory is introduced to overcome the problem of inability to acquire enough valid data to estimate the disruption probability in the COVID-19 context. (3) The independent long-term disruption risks resulting from facilities and transportation failures in more than one level of SCN simultaneously have not been explicitly captured in existing studies. Besides, the single assignment constraints are still common in some SCND studies, which reduces difficulties in solving model but imposes additional restrictions. In this study, we propose the ISN design model characterised by providing each demand market with a pair of supply routes, in which we remove the single assignment constraints and consider the independent disruption of both facility and transportation in multi-level networks. (4) The structural dynamics of ISNs has not been largely explored in existing studies. In this study, we propose a novel flexible redundancy optimisation approach based on ISN structural dynamics, which can effectively alleviate the survivability risks and economic losses resulting from delayed responses under a changing environment.

Problem description
The me-ISN consists of 35 interconnected SCs which, in their integrity, secure the medical equipment provision of southern China's medical demand market. We simplify the complex structure of me-ISN into a three-level intertwined supply network including suppliers, focal firms, and demand markets. The suppliers are mainly in the USA, Canada, and Cambodia, while the focal firms are located in Wenzhou, Hangzhou, and Fujian. The demand markets consist of small and medium-sized medical institutions in southern China. The outbreak and dispersion of COVID-19 resulted in a drastic increase of medical equipment demand. However, the material shortage and transportation links disruptions due to border closures and quarantines makes the survivability of the market and SCs an urgent question. Unlike instantaneous disruption, the impact of the COVID-19 pandemic will not disappear in a short period of time and may radically change operational conditions over the long term, which makes the me-ISN experience long-term and severe uncertainty of current and future conditions El Baz and Ruel 2021;Yang et al. 2020). At the global scale, different degrees of epidemic waves may happen at anytime and anywhere. The disruption caused by the pandemic will bring unpredictable and uncertain risks to the ISN system and affect operation at different levels of the supply network at any time. For the ISNs, it is of great significance to mitigate the risk caused by disruptions and improve the ability to survive in a dynamic environment.
Generally, in the me-ISN case, medical equipment is transported from suppliers to focal firms and then focal firms dispatch those goods to each demand market. The customers in each demand market cannot acquire goods directly from the suppliers. Instead, customers make orders online and focal firms can acquire customer demand information in advance and organise distribution accordingly. As suppliers are distributed worldwide, cross-border logistics is crucial for the me-ISN. However, the facility and transportation links are prone to failure due to the border closures and quarantines due to COVID-19, which may significantly affect the production and delivery of goods, bringing uncertainty risks to both the SCs and demand markets.
The disruption probability of facility and transportation links is treated as the expected value of a fuzzy random variable. Appendix A provides an introduction of fuzzy random theory and explains the estimation process. Notably, the capacity constraints of facility and transportation links are not considered here, but the concept of 'disruption' in the model is an aggregate state of all possible factors that prevent the normal operation of facility and transportation links, such as the damage to the transportation infrastructure (transportation delays), and the unavailability of human or machine resources (manufacturing delays).
The purpose of this model is to optimise the structure of me-ISN, including the selection of supply routes and the assignment of customers under COVID-19 to ensure me-ISN's viability, while minimising the expected total cost in response to the changing environment. A pair of supply routes are assigned to each demand market, which connect the entities at different levels of the me-ISN. The sufficient and necessary condition for a supply route to be selected is that all facility and transportation links operate normally. Customers are mainly supplied through the main routes, while the backup routes only serve the customers when the main supply routes fail.

Adaptive mechanism
As there are significant differences between COVID-19 and common disruptive events (single, discrete, and unique events) in terms of duration, complexity, and influence scale, it is nearly impossible to predict all possible future disruptions ). Decision makers can only estimate the occurrence of these disruptions over a certain period in the future (i.e. planning horizon), which is usually set as 1-3 months (the duration of each time period is not required to be equal), and then optimise/design a structure of me-ISN that can ensure viability at an optimal cost based on the observation of the external environment and the estimation of disruptions.
As pointed out by Ivanov (2020b) and Ruel et al. (2021), viability can be defined as the ability to maintain itself and survive in a changing environment over a long period of time through redesigning structures and re-planning economic performance with longterm impacts. Therefore, the redundancy optimisation approach proposed in this study emphasises the adaptive mechanism, that is, to dynamically adjust and reconstruct the ISN structure in response to the changing environment based on the observation of the external environment and the estimation of the impacts resulting from disruptions.
The flow chart of the adaptive mechanism is shown in Figure 1.
As decision makers face different environment in different time periods, they will set different expected degrees for viability performance. Due to the fact that the external environment is continuously changing, the optimal structure of the me-ISN in period t may be no longer suitable in the t + 1 period. The condition of triggering the adaptive mechanism is that the current viability performance cannot reach a predetermined degree, which implies there is a conflict between the current structure and the external environment. On reaching the trigger point, decision makers should update the model parameters, especially the market demand and the probability of disruption occurrence, and rerun the model to obtain the optimal design under the new situations.

Model formulation
The ISND model is proposed to optimise the me-ISN's structure through a trade-off between total expected operation cost and viability performance. The total cost in the model includes fixed cost, serving cost, shortage cost for the unsatisfied demand of the entire market, and restoration cost for the unselected facilities to maintain and restore critical functions. We present the notations of the model in Table 2.
The components of total expected costs and the corresponding explanation are as follows: (1) Serving cost  Serving cost for demand market k acquiring products through b and u, including procurement and transportation costs.
Fixed costs of opening suppliers i and b, respectively. f j , f u Fixed costs of opening focal firms j and u, respectively.
Failure probabilities of suppliers i and b, respectively. q j , q u Failure probabilities of focal firms j and u, respectively. v ij , v jk Failure probabilities of links ij and jk, respectively. v bu , v uk Failure probabilities of links bu and uk, respectively. p ijk↑ Probability that main supply route ijk works normally. p ijk↓buk↑ Probability that main supply route ijk fails but backup supply route buk works normally. p ijk↓buk↓ Probability that both main supply route ijk and backup supply route buk fail. c pu Shortage cost for each unsatisfied order of the entire market. c re(i) , c re(b) c re(j) , c re (u) Restoration costs for the unselected suppliers i and b and unselected focal firms j and u, respectively. d k Customers' demand in markets k. e ijk , e buk Transportation distance of main supply route ijk and backup supply route buk, respectively (great-circle distance measurement is used.) Optimal results of facility selection of supplier i and focal firms j in each previous period, t ∈ T, b ∈ I, u ∈ J, ij = bu, respectively.
Initial inventories of suppliers i and b, respectively.
Production rates of suppliers i and b, respectively.
Production start time of supplier i and b, respectively.
Production end time of supplier i and b, respectively. R Predetermined number of supply routes, set by the decision-makers and used to limit the total number of routes. α Maximum allowable error coefficient. Decision variables X ijk X ijk = 1, if supplier i and focal firm j are assigned to demand market k as main supply route; 0, for otherwise. X buk X buk = 1, if supplier b and focal firm u are assigned to demand market k as backup supply route; 0, for otherwise. S ijk , S buk Quantity of products shipping to demand market k through main supply route ijk and backup supply routebuk, respectively. Y i , M j 1 if supplier i and focal firm j are selected, and 0 otherwise. Y b , M u 1 if supplier band focal firm u are selected, and 0 otherwise.
In formula (1a), the first term represents the serving cost when customers are served by main supply route ijk, and the second term represents the serving cost when the main supply route fails and the customers are served by backup supply route buk. The serving cost refers to the cost of acquiring and transporting products from the supplier to the customers, which consists of transportation cost and procurement cost. The transportation cost is related to the distance between the facilities, and the procurement cost usually fluctuates due to price volatility (Li et al. 2018;Popescu and Seshadri 2013;Van Mieghem 2003). As the suppliers and focal firms in the me-ISN adopt index-based contracts, the procurement price for each medical equipment order is calculated as pr = ρu + (1 − ρ)s + m, where s, u denote the spot and forward prices for medical equipment, respectively. In particular, s ∼ N[u, σ 2 ], m denotes the unit profit margin for the manufacturers, and ρ denotes the weight of forward price, which may be either above or below 0. Evidently, ρ does not change the expected value of the contract price (i.e. E(pr) = u + m).
(1) Fixed cost Formula (1b) represents the opening and operation cost when a facility is selected to provide products for a demand market, which is related to the geographical location, scale, and reliability to withstand disruptions.
(1) Shortage cost Formula (1c) represents the economic loss resulting from unmet demand of the entire market. The higher the shortage cost is, the lower the service level of the me-ISN. Here, c pu denotes the shortage cost for each unsatisfied order, which is usually predetermined in procurement contracts.
(1) Restoration cost Formula (1d) is the restoration cost for those unselected facilities in me-ISN to maintain or restore critical functions, which are usually related to the severity and scale of disruptions. The restoration of key functions of a facility in the current period may help to reduce the disruption risks in the next period and improve the ISN adaptability.
In summary, the ISND model under facility and transportation disruption risks is formulated as follows: subject to: i j M u are all binary, while S ijk and S buk are continuous on [0, d k ]. Constraints (1g) and (1h) indicate that no supply route can serve as both the main and backup supply route for the same demand market. Additionally, (1g) and (1h) also require that a supply route be selected only when all facilities are installed and the transportation links operate normally. Constraints (1i) and (1j) enforce that each demand market must be served by a pair of supply routes. Constraint (1k) restricts the number of supply routes to meet the requirements set by decision-makers. Constraints (1l) and (1m) enforce that the service level and long-term average survival rate should be no less than a certain level usually predetermined by the decision makers in different periods. Constraints (1n) and (1o) specify that, for each demand market k, the quantity of products shipped through main supply route ijk or backup supply route buk should not exceed the total amount of initial and newly produced inventory of suppliers.
The main implication of the ISND model is optimising the structure to provide flexible redundancy under a facility and transportation disruption through a trade-off between viability performance and total cost.

Combination of facility and transportation disruption
A novel redundancy optimisation approach has been applied to the ISND model, which considers the combination of facility and transportation disruptions. A supply route can only be selected when the facility and connection routes operate normally. Customers are primarily served by the main supply route and, when the main supply route fails, customers will be reassigned to the backup supply route. Figure 2 shows the combinations of main and backup supply routes, the me-ISN can be regarded as providing customers with a combination of main and backup supply routes when it satisfies one of the following three situations. Figure 2(a-c) include two suppliers and one focal firm, one supplier and two focal firms, and two suppliers and two focal firms, respectively. We assume that facilities and transportation disrupt with independent probabilities. The sufficient and necessary condition for a route to be selected is that all facilities and transportation links operate normally, that is, . p ijk↓buk↑ , which indicates the probability that main route ijk fails but backup route buk works normally, is calculated as follows under different situations.
For situation (c), . Furthermore, p ijk↓buk↓ under different situations can be calculated as follows.

Model linearisation
The ISND model is first formulated as an uncertain integer non-linear program. To reduce solution time and difficulties, we analyse the properties of the ISND model and simplify it equivalently. In the model, decision variables X ijk and X buk (i, b ∈ I, j, u ∈ J, ij = bu) are used to indicate three situations: (1) demand market k acquires goods through main supply route ijk; (2) demand market k acquires goods through backup supply route buk when the main supply route fails; (3) no routes provide service for demand market k when the main and backup routes both fail. To simplify the model, we introduce a new decision variable, ϕ on , where ϕ on = 1 if demand market k acquires goods from ijk or buk and ϕ on = 0 if both ijk and buk fail. When ϕ on = 1, we define parameters c on−ijk and c on−buk , which denote the cost of serving demand market k through ijk and buk under disruption, wherec on−ijk = p ijk↑ c ijk and c on−buk = p ijk↓buk↑ c buk .
The ISND model can be first simplified as follows: However, in the simplified ISND model, ϕ on S ijk and ϕ on S buk are still non-linear terms, where ϕ on ∈ {0, 1} and S ijk and S buk are both decision variables that are continuous on [0, d k ], indicating the quantity of medical products shipped to demand market k through supply routes ijk and buk. Generally, as S ijk = f r ijk d k and S buk = f r buk d k , f r ijk and f r buk are continuous variables on [0, 1]. ϕ on S ijk and ϕ on S buk can be rewritten as ϕ on f r ijk d k and ϕ on f r buk d k , and let OK ijk = ϕ on f r ijk , OK buk = ϕ on f r buk . We then apply the linearisation technique proposed by Sherali and Alameddine (1992) to transform them as follows: Similarly, OK buk ≤ f r buk , (2f) Based on the above-mentioned techniques, we can linearise them to obtain an equivalent mixed integer linear model. The linearised ISND model under facility and transportation disruption risks is formulated as follows: subject to:

2(a) − (2h).
By exploring the properties of the model, the optimal structure of the me-ISN can be obtained through a tradeoff between operation cost and viability and there are two main factors affecting the objective value, which are the degree of viability and the number of supply routes. In this study, the optimal numbers of routes and the minimum degree of viability are predetermined by constraints (4d)-(4i). Therefore, in the process of solving the ISND model, we first determine the optimal locations of facilities to form supply routes and then adjust the sizes and combinations of routes according to the constraints and limitations proposed by decision makers.
The ISND model belongs to complex SCND problem, including the facility location and routes allocation problems. According to the mainstream literature, the solutions to these problems can be divided into three categories: (1) Using valid inequalities to strengthen the formulations (Landete and Marín 2009;Aardal et al. 1996), which usually lead to large scale models; (2) Using approximate algorithm (Barros and Labbé 1994;Gao and Robinson 1992); and (3) Using heuristics and meta-heuristics algorithms to obtain near-optimal values (Bumb 2001). Although the linearised model can be solved by general solvers, the solution time is extremely long. Therefore, in Sections 4 and 5, we solve the ISND model by the Lagrangian relaxation and the cellular genetic algorithms, respectively, and then compare the performances of the two algorithms.

Lagrangian relaxation algorithm combined with sub-gradient method
The LR algorithm is an iterative procedure that relaxes the complex constraints to find the upper and lower bounds of the original problem. In each iteration, the Lagrange multipliers (penalties on the relaxed constraints) are updated according to the sub-gradient method, which results in updating the upper and lower bounds. The LR algorithm is iterated until a feasible solution with the desired tolerance is obtained or iteration count t reaches t max .
A typical LR algorithm includes three phases: lower bounding, upper bounding, and updating. The goal is to reduce the gap between the lower and upper bounds as much as possible until the algorithm converges to the global optimum.
The programming logic of LR algorithm is as follows (Figure 3).
By relaxing constraints (4d) and (4e), we constitute the LR of the master problem (the lower bounding phase) as subject to:

Finding the lower bound
To obtain the lower bound of the ISND model, constraints (4d) and (4e) are relaxed. We decompose Z LR (μ, λ) into two subproblems, where Z LB1 indicates the facility location problem and Z LB2 the supply route selection and assignment and product allocation problem.
subject to (5b)-(5f) and (2a)-(2h). It is obvious that Z LB1 is easier to solve, as decision variables M j , Y i in Z LB1 are all binary variables. For given Lagrange multipliers μ and λ, the optimal solution for Z LB1 is given by Since we obtain optimal facility selection Y i * , M j * , the remaining problem, Z LB2 , is the selection and assignment of supply routes for each demand market and the determination of shipment quantity based on the disruption probability to minimise the serving and penalty costs. The optimal objective value of the lower bound in this iteration is denoted as LB(t), LB(t) = min Z LB1 + min Z LB2 , and the optimal solution for the lower bound is denoted as Y * .

Finding the upper bound
To find the upper bound of the ISND model, we use the following heuristic algorithm to construct a feasible solution for L'-ISND using the lower bound solution. We retain the optimal decision of Z LR (μ, λ) and calculate the upper bound based on linearised model L'-ISND. Since Y i * and M j * have been determined, the remaining upper bound subproblem, SPUB, represents the selection and assignment of supply routes and the determination of optimal shipment quantity for each market.
subject to (5b)-(5f) and (2a)-(2h). After obtaining optimal objective value SPUB(Y * ), we remove the facilities that serve no customers and update optimal facility selection Y i * * , M j * * in subproblem SPUB. Then, we calculate the optimal objective value of upper bound in this iteration, * * (f i − c re(i) ) + M j * * (f j − c re(j) )+(c re(i) +c re(j) )], and the optimal solution of the upper bound is denoted asȲ * .

Updating upper and lower bounds
The sub-gradient method is used to update the algorithm parameters (Fisher 2004). With the algorithm iterating, the Lagrange multipliers, the step-sizes update, and the upper and lower bounds are determined to calculate the gap. The algorithm parameters are updated through the following equation: ξ is a fixed step-size parameter and will be halved every few iterations without an improvement in UB and LB. bestUB denotes the best known upper bound and LB(t) the lower bound for current iteration t.
The values of the Lagrange multipliers are updated according to the following rules: We repeat the process of finding UB and LB until a feasible solution with the desired tolerance is obtained, or the minimum value of the step-size is reached. The optimality gap of the algorithm at each iteration t, which is one of the algorithm's stopping criteria, is calculated as follows: When the Lagrangian procedure terminates, if the best known lower bound is equal to the best known upper bound (within some pre-specified tolerance), we have found the optimal solution to the ISND problem. Otherwise, a branch-and-bound algorithm is used to close the gap.

Cellular genetic algorithm
The genetic algorithm is a typical meta-heuristic algorithm that has increasingly matured and been widely used in recent years. Its computational model simulates the natural selection of Darwin's evolutionism, including the biological evolution of the heritage mechanism and the algorithm that continuously chooses optimum individuals. Brown and Sumichrast (2005) noted that, in practice, the genetic algorithm can better address NPhard problems with higher accuracy and shorter calculation times (Kumar and Shanker 2000;Syarif, Yun, and Gen 2002).

Fitness function
In practice, we do not need to consider the inherent properties of the model or problem in searching the optimal solution when using the genetic algorithm, as it can deal with any forms of objective functions and constraints, whether linear or nonlinear, discrete or continuous. Therefore, we introduce the ISND model directly into the genetic algorithm as a fitness function: subject to (1f)-(1o). P is the penalty function used to punish those abnormal results that do not satisfy constraints. A common penalty practice is to assign a large value to those abnormal chromosomes, so that it does not become the final valid solution to be output or prevent it from participating in the next round of computing. In this study, constraints (1f) and (1i)-(1o) can be easily manipulated and implemented in the process of generating chromosomes. Since constraints (1g) and (1h) are more crucial, we add them into the model as penalty functions, being expressed as follows: L is a large real number and P 1 , P 2 are both 0 if the constraints are satisfied; otherwise, the result will be regarded as an invalid solution. δ t in the fitness function is a very small number that decreases with generation t and has two functions. One is to give the worst individuals a chance to reproduce to increase population diversity. The other is to adjust selection pressure, so that δ t is larger at the initial iteration stage, selection pressure is small, and the differences in the individual selection probabilities are insignificant. In the later stage, selection pressure increases and the algorithm soon becomes convergent (Yao, Shi, and Liu 2020).

Cellular genetic operators
As mentioned above, there are three combinations of supply routes, and the facility opening and customer assignments decisions vary under different scenarios. Therefore, we integrate the cellular strategy into the classic genetic algorithm. In the cellular genetic algorithm, individuals are conceptually arranged in a specific topology and only interact with neighbours. As a result, the overlapped small neighbourhoods devote to explore the search space, and a kind of exploration happens owing to the slow spread of individuals. The exploration caused by the genetic operation provides the ability to maintain a proper balance between diversity and convergence (Zhang et al. 2015). The main steps of cellular genetic algorithm are as follows: Step 1: Initialising population and generating cellular states (main, backup, or no supply route). Each population corresponds to a |(I + J) * K| chromosome matrix. Each chromosome is composed of two parts, where the first |I| genes represent the selection of suppliers and the medical equipment shipping quantity and the second |J| genes represent the selection of focal firms for each demand market k. From Figure 4, the populations are placed in a honey-comb grid. Each grid node indicates an individual represented by a unique number. Individuals can only interact with their neighbour.
Step 2: Manipulating the genetic operations to individuals, including crossover, mutation and inversion, and evaluation.
Step 3: Roulette and elite strategy are used to select chromosomes and preserve the best chromosome in each generation t.
Step 4: If f (t) < MAXGEN, outputting the current solution as the optimal solution.
The programming logic of the cellular genetic algorithm is as follows (Figure 5).
Genetic operations include crossover, mutation and inversion, and evaluation, which are primarily used to optimise the sequence of chromosomes and determine whether they are selected. These three genetic operators are applied to the population to form new offspring until stop conditions are reached.
(1) Crossover: The crossover occurs by exchanging two parents' information to provide a powerful exploration capability. We employ a one-cut point crossover operator that randomly selects a cut-point and exchanges the optimum parts from two parents to generate an offspring. We begin by defining a parameter P c of an evolutionary system as the crossover probability.
(2) Inversion and mutation: Modifying one or more gene values from an existing individual results in mutation, which creates a new individual to increase the population's variability. The mutation operator's function is to randomly select some parents and alter some of their genes with a probability equal to the mutation rate. We begin by defining parameter P m of an evolutionary system as the probability for mutation and make the number of mutated genes decrease with each iteration. (3) Evaluation: In nature, it is necessary to provide a driving mechanism for better individuals to survive.  Our evaluation involves associating each chromosome with a fitness value that demonstrates its worth based on the achievement of the objective function.
The higher an individual's fitness value is, the higher are its chances for next generation survival. Therefore, individuals are selected to be parents in the next generation according to their fitness value. After obtaining all fitness values for the chromosomes, a roulette-wheel approach is adopted in the selection procedure.

Numerical study
Here, we test the performance of the ISND model and the corresponding algorithms in terms of optimising the ISN structure, improving flexible redundancy, and ensuring viability at a minimum cost. Moreover, we try to explain the relationship between ISN structure and the achievement of viability.

Data description
We collected operation data during the COVID-19 pandemic from the me-ISN. As the pandemic spread and quarantine policies kept adjusting, the me-ISN went through four typical periods, denoted as scenarios A, B, C, and D. The available facilities, transportation links and order size of the demand markets under scenarios A-D are summarised in Table 3. Under different scenarios, the facilities and transportation links suffered independent disruption risks, which were quite different in scale, complexity, severity, and duration. The disruption levels (DILs) of the different facilities and transportation links under different scenarios, along with their means and standard deviations, are also shown in Table 3. The disruption level of each scenario is treated as the expected value of the fuzzy random number, which can be estimated by experts at the beginning of each period based on the real observed data with the help of a distributed group support system (GSS) and calculated using the algorithm presented in Appendix A.
To ensure the survivability of the me-ISN and secure the medical equipment provision in southern China, we optimise me-ISN's structure to create flexible redundancy dynamically. From Section 3.2, the structure of the me-ISN needs to be adjusted dynamically based on the observation of the external environment and the estimation of the disruption occurrence. Therefore, at the beginning of each period, decision-makers are required to provide a predetermined degree of viability performance, including current service level and average survival rate, as well as a corresponding trigger point as the alarming signal of structure redesign, which are shown in Table 3. The predetermined degree of viability performance represents the optimal level, while the corresponding trigger point represents the minimum level that decision makers can accept. If the observed viability performance value is lower than the minimum level, it is a signal to redesign the structure.
In the numerical study, the serving cost (c ijk ) of supplying demand market k through suppliers i and focal firms j is calculated as: c ijk = 1000S ijk + 5e ijk . The opening and operation cost (f i , f j ) of facilities and restoration cost (c re ) are fixed under different scenarios, related to their geographical locations, scales, and disruption levels. The shortage cost (c pu ) is a fixed parameter, c pu = 800 for scenarios A and B and c pu = 1000for scenarios C and D.
We adopt the Gurobi MIP solver, Lagrangian relaxation algorithm, and cellular genetic algorithm to solve the ISND problem using me-ISN data. All computations are implemented in MATLAB R2019b on a Win64 laptop with core TM i7-8656U CUP @ 2.40 GHZ Intel processor with 16 GB RAM. The computation time is limited to 36,000 s. The parameters of the Lagrangian relaxation and cellular genetic algorithms are shown in Table 4.

Analysis of the performance of different solvers and algorithms
The computation results are shown in Table 5, where Gap_1 and Gap_2 represent the gaps between the upper and lower bounds of Gurobi solver and LR algorithm, respectively, while CPU time represents the computation time of different solvers and algorithms. To test the robustness of the above three algorithms and solver, we run datasets A and B (scenario A and B) 100 times, and datasets C and D 50 times. The results reported in Table 5 are the average optimal values of objective. These robustness tests show that the above three algorithms and solver have strong stability and robustness, and the gaps between the upper and lower bounds are below the optimality tolerance (0.01). For datasets A and B, the standard deviation of the 100 times operation is 0.0132 and 0.0316, while for datasets C and D the standard deviation of 50 times operation is 0.0371 and 0.0591. Although the standard deviation increases with dataset size, it is still in a permissible range. From the calculation results, with the increases in maximum supply routes (R) and disruption level, the objective values keep increasing, indicating that both the network structure and disruption level influence the network design results.
By comparing the performances of the different algorithms and solvers, it is obvious that the Gurobi solver and LR algorithm are more suitable for scenarios A and B. They can obtain the exact solutions for small and medium scale problems within a short time. However, when the problem scale increases, the calculation time and gaps of the Gurobi solver and LR algorithm increase significantly, and the accuracy of the solution decreases. The cellular genetic algorithm performs better than the Gurobi solver and LR algorithm for scenarios C and D and the calculation time is shorter. In summary, the Gurobi solver and LR algorithm are more suitable for small and medium scale ISND problems, while the accuracy of solution decreases and the calculation time increases when solving larger scale problem. The cellular genetic algorithm can solve ISND problems of different scales and obtain near-optimal solutions with high accuracy and short computation time. Therefore, in practice, we can solve small and medium scale problems by general solvers or the LR algorithm and, for largescale problems that require to be solved in given time, the cellular genetic algorithm is recommended. Since the purpose of the ISND model is to dynamically optimise redundancy, the cellular genetic algorithm can be used to provide solution in response to a rapidly changing environment.

Analysis of the impact of disruption level and maximum supply routes (R) on penalty cost
From the results in Table 5, intuitively, the total cost of ISN design, which consists of fixed cost, serving cost, and penalty costs (shortage cost for the unsatisfied demand and restoration cost for unselected facilities to maintain and restore critical functions), always goes up with the increase in disruption levels and R. However, an encouraging observation is that the penalty cost goes down with the increase in R. The relationship between the penalty cost and R under different disruption levels is shown in Figure 6.
From Figure 6, for each scenario, under the same disruption level, the increase in R will lead to a decrease in penalty cost. The larger the ISN scale is, the more significant the decrease becomes. Moreover, with the increase in R, the decrease in penalty cost becomes more obvious for high disruption level situations, compared with low disruption level situations. For example, under scenario A, the penalty cost decreases by 28.488% for a high disruption level (1.25DIL) and 21.401% for a low disruption level (0.75DIL); under scenario D, the penalty cost decreases by 73.394% for a high disruption level (1.25DIL) and 68.581% for a low disruption level (0.75DIL). This observation shows the effectiveness of deploying more supply routes in mitigating the negative impacts resulting from disruptions. With the increase in disruption level, more supply routes should be installed if possible.

Analysis on the impact of maximum supply routes (R) on viability performance
To explain the above observation, we present the relationship between R and viability performance of the me-ISN under different disruption levels in Figure 7 and explain why the increase of R will lead to a decrease in penalty cost. The performance of viability includes three aspects: (1) service level, (2) current ISN adaptability, and (3) long-term average survival rate of the ISN, which are equally important in the measurement of viability performance in this numerical study. The current ISN adaptability can be represented by the current SC survival rate. If the facilities of an SC are assigned to at least one demand market, this SC can be seen as survival. We use the following equation to aggregate the three aspects: viability performance = service level 2 + current adaptability 2 +average survival rate 2 .  As is shown in Figure 7, for each scenario, it is evident that the ISN viability performance becomes better with the increase in R. The increase in R makes it possible for the ISN to create more flexible redundancy, which means that the number of survival SCs increases. With the improving of SCs' survival rate, the ISN structure becomes more reliable and its ability of satisfying customers' demand improves, which leads to the decrease in penalty cost. Although the damage and economic loss caused by disruption risks are unavoidable, the negative impacts can be alleviated by applying the novel redundancy optimisation method proposed in this study.
This observation allows us to identify the relationship between the ISN structure and viability performance. What is more interesting is the linear pattern of the increase in viability performance with the rise in R. This finding can be helpful for the decision makers to set the most appropriate number of supply routes according to their predetermined viability performance when designing the me-ISN, which can improve the efficiency of decision process.

Analysis of the trade-off between viability performance and the total cost
The ISND model proposed in this study captures the trade-off between total cost and viability performance. Such a trade-off can be developed by varying the predetermined viability degree (θ 1 , θ 2 ), as shown in Figure 8.
By varying the predetermined viability degree under different scenarios, we find that, under different disruption levels, the total cost always goes up with the increase in predetermined viability degree. The fact that decision makers usually enlarge the network by deploying more supply routes to meet the higher requirement on viability performance, will lead to an increase in total cost. An interesting insight is the linear pattern of the increase in total cost with the rise in predetermined viability degree. This finding can be helpful as it allows a decision-maker to predict the expected total cost when designing ISN with the purpose to improve viability performance.
Focusing on the disruption levels can provide further insights into the relationship between viability performance and total cost. Compared with the total cost change with the increase in viability degree under a low disruption level, the total cost seems less sensitive to the increase in viability degree under a high disruption level. For example, under scenario C, the total cost increases by 80.079% for a low disruption level, but only increases by 66.431% for a high disruption level. Moreover, in specific situations, the increase in viability degree will not lead to substantial increases in total cost. For example, under scenario D, when the disruption is set as 1.25 DIL and when the viability degree increased from (0.875, 0.775) to (0.90, 0.80), there is only a slight increase in total cost. This observation implies that there may be opportunities to improve the viability performance of me-ISN while staying cost efficient under specific disruption scenarios.

Comparison with deterministic scenarios
In Section 6.2, we analyse the computation results of the ISND model under different scenarios, and found the relationship between ISN structures and viability performance. However, the ISND model is designed to solve the complex supply network design problem under continuous uncertainty risks, compared with the deterministic ISND (D-ISND) model without considering disruption risks, whether the optimal network design of ISND model performs better requires further analysis.
In Table 6, we compare the specific performances of the D-ISND and the ISND models under different scenarios. The D-ISND model is constructed based on the deterministic SCN model with the objective of minimising total cost in the absence of facility and transportation disruptions, as proposed by Daskin (1995). The benchmark model is shown in Appendix B.
In Table 6, the proportion of penalty cost to total cost is denoted as P%. The total cost growth rate under the ISND model compared with the corresponding disruption situations under the D-ISND model is denoted as C%. The service level and the long-term average survival rate (AS_rate) under different conditions are also reported. Columns 2-5 represent the objective values, the proportion of penalty cost to total cost, service level, and long-term average survival rate of the D-ISND model under risk-free, low-risk, medium-risk, and highrisk situations, respectively. When disruptions occur, the shortage cost is set to be the maximum serving cost of demand markets acquiring products from suppliers and the restoration cost is set to be the maximum cost to resume critical functions. Columns 6-14 represent the objective values, proportion of penalty cost to total cost, cost growth rate, service level, and long-term average survival rate of the ISND model under different disruption levels and different supply route opening situations, respectively, where '-' indicates that the specified situations are not considered. The advantages of the ISND model are clearly demonstrated in Table 6. As the uncertainty risks of facility and transportation disruptions are not considered under the D-ISND model, losses are relatively severe when disruption occurs. The penalty cost increases significantly with the increase in disruption risks and the viability performance for each scenario can hardly reach the degree predetermined by decision makers. As for the ISND model, although the increase in total cost is unavoidable, the model considering dual impacts of facilities and transportation disruption can significantly alleviate great losses when risks occur and guarantee the number of supply routes opened to ensure ISN viability.
Moreover, for small-scale scenarios, such as scenarios A and B, optimising the ISN structure using the ISND model will bring a higher cost compared with the D-ISND model. However, under large-scale scenarios, such as scenarios C and D, the cost advantage of the ISND model can be observed clearly.

Comparison with other structural redundancy optimisation approaches
As summarised in Section 2.2, multiple resilience methods have been adopted to mitigate the negative impacts of disruptions under different scenarios. Compared with other structural redundancy optimisation approaches, whether the optimal network design of ISND model performs better requires further analysis. To test the advantage of this approach in withstanding long-term unpredictable disruptions, we compare it with the following three resilience methods, which are widely used in current industrial practice: (1) Backup suppliers: Contracting backup suppliers to serve factories when the primary suppliers are not available. Traditionally, the backup suppliers are assumed to be not affected by disruptions and are able to serve demand markets under all scenarios. However, under the situations of this study, both primary and backup suppliers can be affected by disruptions at any time.
(2) Facility fortification: Enhancing the resilience of an SCN through investment to improve the facility fortification degree. Examples of such investments are the acquisition, installation, and implementation of infection control measures to contain and prevent diseases from disabling a workface. (3) Multiple transportation channels: Providing additional capacities for transportation links (i.e. contracting with third-party logistics providers (3PLs)), since production and distribution facilities as well as transportation links are vulnerable to disruption risks.
Dataset D (scenario D) is used to compare performance in terms of total cost and viability performance of the above-mentioned three resilience methods as well as the method proposed in this study. The predetermined viability degree is (0.80, 0.70). To test the ability of withstanding continuous uncertainty risks of the four network structures, we add disruption risks incrementally from 1DIL to 2DIL. Moreover, we set the trigger point at (0.65, 0.55), which serves as a signal when the network structure is no longer adaptive to the current environment. For all mentioned methods, the later the trigger point is reached, the better the viability performance of the network structure is. From Figure 9, when the disruption level is set as 1DIL, the cost advantage of the other three resilience methods is obvious. However, with the increase in the disruption level, the cost advantage of the proposed method can be observed gradually. As for the viability performance of the four resilience methods, the advantage of the structure characterised by main and backup supply routes for each demand market is more significant and the trigger point moves backwards. Although the viability performance decreases with the increases in disruption level, it can meet the minimum requirements of decision makers. The redundancy optimisation method proposed in this study is more suitable for long-term unpredictable continuous disruptions, as the flexible redundancy can be provided dynamically to meet the demand of surviving in a changing environment.

Managerial implications
The above-mentioned analysis results demonstrate the following implications from a managerial perspective: (1) The relationship of ISN structure and its viability performance has been observed. This finding indicates that the increase in the reliability of ISN structure will improve the ISN's ability in maintaining SCs survivability and securing the service level.
(2) For decision makers, the trade-off between viability performance and total cost is critical. Generally, the higher the demand in viability performance, the more supply routes need to be installed and the higher the total cost becomes. (3) An interesting insight is that the linear pattern of increase in total cost with the rise in predetermined viability degree, which allows a decision-maker to predict the expected total cost increase when designing ISN to improve viability performance. Besides, the linear pattern of increase in viability performance with rise in R is also helpful, since it allows the decision makers to predict the most appropriate number of supply routes according to their predetermined viability performance, which can improve the efficiency of decision process. (4) Compared to the deterministic ISND model, although sometimes the increase in total cost is unavoidable, the ISND model considering both facilities and transportation disruption can significantly alleviate great losses when disruptions occur and guarantee the number of supply routes opened to ensure ISN viability. (5) Compared to other structural redundancy optimisation methods, the application of the novel redundancy optimisation method proposed here can better mitigate the long-term unpredictable uncertainty risks by a relatively small increase in the design costs.

Conclusions
In this paper, we discuss the intertwined supply network design and redundancy optimisation problem from the viability perspective. Following the views of Ivanov and Dolgui (2020), this paper contributes to the research on ISN viability and supply chain network design under continuous changing environment. The novel flexible redundancy optimisation approach and the adaptive mechanism based on ISN structural dynamics have been proposed in this study, which can effectively alleviate the survivability risk and economic loss resulting from delayed responses under changing environment. The corresponding ISND model which capture the trade-off between viability performance and operation cost, and algorithms can address the ISN design problem with high efficiency, thereby providing managers with a useful decision-making tool. To analyse ISND model's performance in terms of optimising ISN structure, improving the flexible redundancy and ensuring viability in the context of the COVID-19 pandemic, we consider the example of a me-ISN in southern China. From the numerical study, we observe and explain the relationship between ISN structure and viability performance. Moreover, compared to the deterministic ISND model, it is evident that considering uncertainty caused by facility and transportation disruptions in the design phase can reduce economic loss significantly when long-term unpredictable disruptions occur. Compared to other structural redundancy optimisation methods, the application of the novel redundancy optimisation method proposed in this study can better mitigate long-term unpredictable uncertainty risks by a relatively small raise in design costs.
One of the major limitations of this study is that the demand of each market is assumed to be deterministic, as decision makers can acquire demand information online. Although this assumption is suitable for the structure of ISN in this study, whose market consists of medical institutions, it may be not appropriate for ISNs that provide products and service for individual customers, whose demand may be highly affected by disruptions and hard to acquire in advance. Further, as it is nearly impossible to predict all possible future disruptions, the design of the ISN structure has been formulated as a single-period problem, which implies that some parameters in the model should be updated at the beginning of each planning horizon to improve decision-making accuracy, but may lead to extra transformation cost. We use the restoration cost to formulate the necessary cost in the process of structure transformation. However, the complex forms of penalty such as loss of orders and distribution delay deserve more attentions under more complex situations. Moreover, the inventory strategies are overlooked in this study. Although this is a common assumption in SCND model, it may be no longer applicable for complex network design problems which are sensitive to inventory strategies.
These limitations provide opportunities for future research. For example, integrating the inventory strategy in the network structure design problem is necessary to be considered in future studies to ensure viability. Moreover, in this study, the disruption probabilities of facilities and routes are treated as fuzzy random variables, which can be estimated by experts when historical data are imperfect. We will try to explore more methods to estimate the probability of disruption occurrence when historical data on disruptions are not enough or less informative. Finally, the solution algorithms have the further optimisation space. In future studies, we will optimise the solution algorithms and even study other intelligent algorithms to improve the solution efficiency.